536
11 Biomolecular Structure and Interactions
over a range of frequencies. The light absorption frequencies observed come from
the resonant absorption at or near the normal modes of vibration for the system.
Molecules have a set of normal modes of vibration. The most general internal
vibratory motion of the molecule will be a sum of normal vibrational modes.
Vibrational modes of molecules are determined by the structure of the molecules
and, to a lesser degree, by the interactions with nearby molecules. Vibrational modes
involving stretching of the interatomic distance tend to have higher frequencies than
distortions which leave the interatomic distance fixed, and so the spectral wavenumbers for IR resonant absorption due to stretching modes are above those for
modes with lateral shifts of atoms and side chains (with those above about 1600/cm
likely to be stretching mode absorptions). In the IR wave-number range 480–
4000/cm, the resonant absorption is mostly from the ground state to the first excited
state if either state has an electric dipole moment. The stretching mode vibration of
the first excitation is usually at an energy far below the the bond-breaking energy,
so the vibration is close to that of a harmonic oscillator.
The maximum number of distinct normal modes is determined by the number
of vibrational degrees of freedom available to the molecule, i.e. the number of
independent coordinates needed to specify the internal motions of the system. For a
system with N particles in a molecule capable of motion, there will be 3N possible
motions. However, of these, 3 will be simple uniform translations of the whole
molecule through space and there are 2 or 3 more to specify the way the molecule
is rotating as a whole. If the molecule has axial symmetry, rotation around the
symmetry axis may not be detectable. 17 In this case, only two angular velocities
are needed in determining the energy absorbed by rotation of the molecule.
If the three coordinates of the center of mass of the molecule, X, Y, Z are used
to track the translational motion, we can eliminate the translational motion from the
possible motions by selecting a center-of-mass coordinate system. In this system of
coordinates, there will no longer be a translational motion of the molecule and so
this motion will no longer contribute to the kinetic energy of the atoms.
We are free to select the direction of the x, y, z axes. We are also free to select
three angles which determine the orientation of a molecule, as this choice should not
affect how the molecule executes internal vibrations. If the molecule has a symmetry
axis, rotations with small angular velocities around that axis are not observable, so
that one cannot eliminate a degree of freedom by fixing the angle of rotation around
this symmetry axis. This leaves 3N − 6 independent internal motions for a nonaxial molecule, and 3N − 5 if there is axial symmetry. Distinct vibrational modes
correspond to the symmetry group of the molecule.
A water molecule, considered as a three particle system, has no rotational axis
of symmetry, so 3N − 6 = 3. It can vibrate in three different normal modes:
bending, with the lowest wave number 1595 cm −1 , symmetric stretch, with the
lowest wave number 3657 cm −1 , and an antisymmetric, with the lowest wave
number 3756 cm −1 . Carbon dioxide, having three atoms all in a line and so an axis
17 For fast rotations, the molecule may expand out from the axis, making such motion detectable.
11 Biomolecular Structure and Interactions
over a range of frequencies. The light absorption frequencies observed come from
the resonant absorption at or near the normal modes of vibration for the system.
Molecules have a set of normal modes of vibration. The most general internal
vibratory motion of the molecule will be a sum of normal vibrational modes.
Vibrational modes of molecules are determined by the structure of the molecules
and, to a lesser degree, by the interactions with nearby molecules. Vibrational modes
involving stretching of the interatomic distance tend to have higher frequencies than
distortions which leave the interatomic distance fixed, and so the spectral wavenumbers for IR resonant absorption due to stretching modes are above those for
modes with lateral shifts of atoms and side chains (with those above about 1600/cm
likely to be stretching mode absorptions). In the IR wave-number range 480–
4000/cm, the resonant absorption is mostly from the ground state to the first excited
state if either state has an electric dipole moment. The stretching mode vibration of
the first excitation is usually at an energy far below the the bond-breaking energy,
so the vibration is close to that of a harmonic oscillator.
The maximum number of distinct normal modes is determined by the number
of vibrational degrees of freedom available to the molecule, i.e. the number of
independent coordinates needed to specify the internal motions of the system. For a
system with N particles in a molecule capable of motion, there will be 3N possible
motions. However, of these, 3 will be simple uniform translations of the whole
molecule through space and there are 2 or 3 more to specify the way the molecule
is rotating as a whole. If the molecule has axial symmetry, rotation around the
symmetry axis may not be detectable. 17 In this case, only two angular velocities
are needed in determining the energy absorbed by rotation of the molecule.
If the three coordinates of the center of mass of the molecule, X, Y, Z are used
to track the translational motion, we can eliminate the translational motion from the
possible motions by selecting a center-of-mass coordinate system. In this system of
coordinates, there will no longer be a translational motion of the molecule and so
this motion will no longer contribute to the kinetic energy of the atoms.
We are free to select the direction of the x, y, z axes. We are also free to select
three angles which determine the orientation of a molecule, as this choice should not
affect how the molecule executes internal vibrations. If the molecule has a symmetry
axis, rotations with small angular velocities around that axis are not observable, so
that one cannot eliminate a degree of freedom by fixing the angle of rotation around
this symmetry axis. This leaves 3N − 6 independent internal motions for a nonaxial molecule, and 3N − 5 if there is axial symmetry. Distinct vibrational modes
correspond to the symmetry group of the molecule.
A water molecule, considered as a three particle system, has no rotational axis
of symmetry, so 3N − 6 = 3. It can vibrate in three different normal modes:
bending, with the lowest wave number 1595 cm −1 , symmetric stretch, with the
lowest wave number 3657 cm −1 , and an antisymmetric, with the lowest wave
number 3756 cm −1 . Carbon dioxide, having three atoms all in a line and so an axis
17 For fast rotations, the molecule may expand out from the axis, making such motion detectable.